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Lecture9-new

# Lecture9-new - 6.254 Game Theory with Engineering...

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6.254 : Game Theory with Engineering Applications Lecture 9: Computation of NE in finite games Asu Ozdaglar MIT March 4, 2010 1

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Game Theory: Lecture 9 Introduction Introduction In this lecture, we study various approaches for the computation of mixed Nash equilibrium for finite games. Our focus will mainly be on two player finite games (i.e., bimatrix games). We will also mention extensions to games with multiple players and continuous strategy spaces at the end. The two survey papers [von Stengel 02] and [McKelvey and McLennan 96] provide good references for this topic. 2
Game Theory: Lecture 9 Zero-Sum Finite Games Zero-Sum Finite Games We consider a zero-sum game where we have two players. Assume that player 1 has n actions and player 2 has m actions. We denote the n × m payoff matrices of player 1 and 2 by A and B . Let x denote the mixed strategy of player 1, i.e., x X , where X = { x | n i = 1 x i = 1, x i 0 } , and y denote the mixed strategy of player 2, i.e., y Y , where Y = { y | m j = 1 y j = 1, y j 0 } . Given a mixed strategy profile ( x , y ) , the payoffs of player 1 and player 2 can be expressed in terms of the payoff matrices as, u 1 ( x , y ) = x T Ay , u 2 ( x , y ) = x T By . 3

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Game Theory: Lecture 9 Zero-Sum Finite Games Zero-Sum Finite Games Recall the definition of a Nash equilibrium: A mixed strategy profile ( x * , y * ) is a mixed strategy Nash equilibrium if and only if ( x * ) T Ay * x T Ay * , for all x X , ( x * ) T By * ( x * ) T By , for all y Y . For zero-sum games, we have B = - A , hence the preceding relation becomes ( x * ) T Ay * ( x * ) T Ay , for all y Y . Combining the preceding, we obtain x T Ay * ( x * ) T Ay * ( x * ) T AY , for all x X , y Y , i.e., ( x * , y * ) is a saddle point of the function x T Ay defined over X × Y . Note that a vector ( x * , y * ) is a saddle point if x * X , y * Y , and sup x X x T Ay * = ( x * ) T Ay * = inf y Y ( x * ) T Ay . (1) 4
Game Theory: Lecture 9 Zero-Sum Finite Games Zero-Sum Finite Games For any function φ : X × Y 7→ R , we have the minimax inequality : sup x X inf y Y φ ( x , y ) inf y Y sup x X φ ( x , y ) , (2) Proof: To see this, for every ¯ x X , write inf y Y φ ( ¯ x , y ) inf y Y sup x X φ ( x , y ) and take the supremum over ¯ x X of the left-hand side. Eq. (1) implies that inf y Y sup x X x T Ay sup x X x T Ay * = ( x * ) T Ay * = inf y Y ( x * ) T Ay sup x X inf y Y x T Ay , which combined with the minimax inequality [cf. Eq. (2)], proves that equality holds throughout in the preceding. Hence, a mixed strategy profile ( x * , y * ) is a Nash equilibrium if and only if ( x * ) T Ay * = inf y Y sup x X x T Ay = sup x X inf y Y x T Ay . We refer to ( x * ) T Ay * as the value of the game . 5

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Game Theory: Lecture 9 Zero-Sum Finite Games Zero-Sum Finite Games We next show that finding the mixed strategy Nash equilibrium strategies and the value of the game can be written as a pair of linear optimization problems.
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